Nonlinear Negotiation Approaches for Complex-Network Optimization: A Study Inspired by Wi-Fi Channel Assignment

At the present time, Wi-Fi networks are everywhere. They operate in unlicensed radio-frequency spectrum bands (divided in channels), which are highly congested. The purpose of this paper is to tackle the problem of channel assignment in Wi-Fi networks. To this end, we have modeled the networks as multilayer graphs, in a way that frequency channel assignment becomes a graph coloring problem. For a high number and variety of scenarios, we have solved the problem with two different automated negotiation techniques: a hill-climber and a simulated annealer. As an upper bound reference for the performance of these two techniques, we have also solved the problem using a particle swarm optimizer. Results show that the annealer negotiator behaves as the best choice because it is able to obtain even better results than the particle swarm optimizer in the most complex scenarios under study, with running times one order of magnitude below. Finally, we study how different properties of the network layout affect to the performance gain that the annealer is able to obtain with respect to the particle swarm optimizer.Comment: This is a pre-print of an article published in Group Decision and Negotiation. The final version is available online at https://doi.org/10.1007/s10726-018-9600-

L(2,1)-labelling of graphs

International audienceAn $L(2,1)$-labelling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geq 2$ if $dist(x,y)=1$ and $|f(x)-f(y)|\geq 1$ if $dist(x,y)=2$, where $dist(u,v)$ is the distance between the two vertices~$u$ and~$v$ in the graph $G$. The \emph{span} of an $L(2,1)$-labelling $f$ is the difference between the largest and the smallest labels used by $f$ plus $1$. In 1992, Griggs and Yeh conjectured that every graph with maximum degree $\Delta\geq 2$ has an $L(2,1)$-labelling with span at most $\Delta^2+1$. We settle this conjecture for $\Delta$ sufficiently large.Un $L(2,1)$-étiquettage d'un graphe est une fonction $f$ de l'ensemble des sommets vers les entiers positifs telle que $|f(x)-f(y)|\geq 2$ si $dist(x,y)=1$ et $|f(x)-f(y)|\geq 1$ si $dist(x,y)=2$, où $dist(u,v)$ est la distance entre les sommets~$u$ et~$v$ dans le graphe $G$. Le \emph{span} d'un $L(2,1)$-étiquettage $f$ est la différence entre la plus grande et la plus petite étiquette utilisée par $f$ plus $1$. En 1992, Griggs et Yeh ont conjecturé que tout graphe de degré maximum $\Delta\geq 2$ a un $L(2,1)$-étiquettage de span au plus $\Delta^2+1$. Nous confirmons cette conjecture pour $\Delta$ suffisamment grand

L(2,1)-labelling of graphs

International audienceAn $L(2,1)$-labelling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geq 2$ if $dist(x,y)=1$ and $|f(x)-f(y)|\geq 1$ if $dist(x,y)=2$, where $dist(u,v)$ is the distance between the two vertices~$u$ and~$v$ in the graph $G$. The \emph{span} of an $L(2,1)$-labelling $f$ is the difference between the largest and the smallest labels used by $f$ plus $1$. In 1992, Griggs and Yeh conjectured that every graph with maximum degree $\Delta\geq 2$ has an $L(2,1)$-labelling with span at most $\Delta^2+1$. We settle this conjecture for $\Delta$ sufficiently large.Un $L(2,1)$-étiquettage d'un graphe est une fonction $f$ de l'ensemble des sommets vers les entiers positifs telle que $|f(x)-f(y)|\geq 2$ si $dist(x,y)=1$ et $|f(x)-f(y)|\geq 1$ si $dist(x,y)=2$, où $dist(u,v)$ est la distance entre les sommets~$u$ et~$v$ dans le graphe $G$. Le \emph{span} d'un $L(2,1)$-étiquettage $f$ est la différence entre la plus grande et la plus petite étiquette utilisée par $f$ plus $1$. En 1992, Griggs et Yeh ont conjecturé que tout graphe de degré maximum $\Delta\geq 2$ a un $L(2,1)$-étiquettage de span au plus $\Delta^2+1$. Nous confirmons cette conjecture pour $\Delta$ suffisamment grand

Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)

We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography, comments are welcome, published as a Dynamic Survey in Electronic Journal of Combinatoric

A Unified Framework for Integer Programming Formulation of Graph Matching Problems

Graph theory has been a powerful tool in solving difficult and complex problems arising in all disciplines. In particular, graph matching is a classical problem in pattern analysis with enormous applications. Many graph problems have been formulated as a mathematical program then solved using exact, heuristic and/or approximated-guaranteed procedures. On the other hand, graph theory has been a powerful tool in visualizing and understanding of complex mathematical programming problems, especially integer programs. Formulating a graph problem as a natural integer program (IP) is often a challenging task. However, an IP formulation of the problem has many advantages. Several researchers have noted the need for natural IP formulation of graph theoretic problems. The aim of the present study is to provide a unified framework for IP formulation of graph matching problems. Although there are many surveys on graph matching problems, however, none is concerned with IP formulation. This paper is the first to provide a comprehensive IP formulation for such problems. The framework includes variety of graph optimization problems in the literature. While these problems have been studied by different research communities, however, the framework presented here helps to bring efforts from different disciplines to tackle such diverse and complex problems. We hope the present study can significantly help to simplify some of difficult problems arising in practice, especially in pattern analysis

Griggs and Yeh's Conjecture and L(p,1)-labelings

International audienceAn L(p,1)-labeling of a graph is a function f from the vertex set to the positive integers such that |f(x) − f(y)| ≥ p if dist(x, y) = 1 and |f(x) − f(y)| ≥ 1 if dist(x, y) = 2, where dist(x,y) is the distance between the two vertices x and y in the graph. The span of an L(p,1)- labeling f is the difference between the largest and the smallest labels used by f. In 1992, Griggs and Yeh conjectured that every graph with maximum degree Δ ≥ 2 has an L(2, 1)-labeling with span at most Δ^2. We settle this conjecture for Δ sufficiently large. More generally, we show that for any positive integer p there exists a constant Δ_p such that every graph with maximum degree Δ ≥ Δ_p has an L(p,1)-labeling with span at most Δ^2. This yields that for each positive integer p, there is an integer C_p such that every graph with maximum degree Δ has an L(p,1)-labeling with span at most Δ^2 + C_p

Graph Labellings with Variable Weights, a Survey

Graph labellings form an important graph theory model for the channel assignment problem. An optimum labelling usually depends on one or more parameters that ensure minimum separations between frequencies assigned to nearby transmitters. The study of spans and of the structure of optimum labellings as functions of such parameters has attracted substantial attention from researchers, leading to the introduction of real number graph labellings and λ-graphs. We survey recent results obtained in this area. The concept of real number graph labellings was introduced a few years ago, and in the sequel, a more general concept of λ-graphs appeared. Though the two concepts are quite new, they are so natural that there are already many results on each. In fact, even some older results fall in this area, but their authors used a different mathematical language to state their achievements. Since many of these results are so recent that they are just appearing in various journals, we would like to offer the reader a single reference for the state of art as well as to draw attention to some older results that fall in this area